The Leavitt Path Algebras of Generalized Cayley Graphs

Let n be a positive integer. For each 0 ≤ j ≤ n - 1 , we let C n j denote Cayley graph for the cyclic group Z n with respect to the subset { 1 , j } . For any such pair ( n , j ), we compute the size of the Grothendieck group of the Leavitt path algebra L K ( C n j ) ; the analysis is related to a c...

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Bibliographic Details
Published inMediterranean journal of mathematics Vol. 13; no. 1; pp. 1 - 27
Main Authors Abrams, Gene, Aranda Pino, Gonzalo
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.02.2016
Subjects
Mathematics
Mathematics and Statistics
Secondary 11B39
05C99
Cayley graph
Primary 16S99
Leavitt path algebra
Fibonacci sequence
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Summary:Let n be a positive integer. For each 0 ≤ j ≤ n - 1 , we let C n j denote Cayley graph for the cyclic group Z n with respect to the subset { 1 , j } . For any such pair ( n , j ), we compute the size of the Grothendieck group of the Leavitt path algebra L K ( C n j ) ; the analysis is related to a collection of integer sequences described by Haselgrove in the 1940s. When j = 0, 1, or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras L K ( C n j ) as the Leavitt path algebras of graphs having at most three vertices. The analysis in the j = 2 case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-014-0464-4